2.5 Energy method
2.5.3 Energy method used in case of dynamic deformation
Selecting deformation character by minimising the external load When a cylindrical tube subjected to an axial load P collapses into an axisymmetric folding pattern (see Section 6.1), its large deformation load- carrying capacity varies with the displacement periodically, but the energy dissipation in a load cycle can be written in the following form
[2.91]
where lis the half length of the fold and Aand Bare coefficients depend- ing on material and geometry. On the other hand, the work done by the external load is
[2.92]
with C being another coefficient. Therefore, equating Eq. [2.91] and Eq. [2.92] gives
[2.93]
with A¢ =A/Cand B¢ =B/C. Since Pgiven in Eq. [2.93] is an upper bound, minimising it in terms of lwill result in
[2.94]
This example shows how the character length in a large deformation mech- anism can be determined by minimising the required load.
To take another example, when a cylindrical tube with longitudinal pre- cracks is compressed in the axial direction, the large deformation mecha- nism involves both bending energy and fracture energy. The latter is proportional to the number of fractures which occur, whilst the former is inversely proportional to this number. Again, an optimum number for the fractures occurring in the tube can be obtained from the energy balance argument with the external load minimised.
where Eindenotes the work done by the force pulse F(t). It should be noted that, although the pulse F(t) is prescribed, the relevant displacement is not prescribed, so that the value of Einis known only when the dynamic response of the structure is solved. Therefore, unlike that in the quasi-static deformation, the energy equation [2.95] in the dynamic case will not lead directly to an explicit expression on the relationship between force Fand displacement D.
In fact, after plastic dissipation has been completed, the kinetic energy K still varies with time, because the remaining energy (Ein-D) still periodi- cally transfers its form between the kinetic energy Kand the elastic strain energy Weduring the final elastic vibration phase of the system. Only when (K +We) <<Ein, can the rigid-plastic idealisation be adopted and Eq. [2.76]
used to assess the final deformation of the system after a dynamic (pulse) loading.
Upper bound of final displacement of structures under impulsive loading
Impulsive loading refers to an intense dynamic load, which only applies to a structure in an infinitesimal time period (at t =0) but results in a finite impulse. This kind of dynamic load gives the whole structure or a part of it an initial velocity distribution at t=0, but no force pulse applies afterwards.
Martin (1964) proved that an upper bound to the final displacement of an impulsively loaded structure can be given by (see Stronge and Yu, 1993, Section 2.4.5)
[2.96]
where Kois the initial kinetic energy of the structure due to the impulsive loading, and Psdenotes the quasi-static collapse force of the structure, while Psand Dfshould be at the same point and along the same direction.
For example, for the previous example shown in Fig. 2.12(a), suppose that instead of a quasi-static loading by the round-headed indenter, the beam is impinged by a round-headed projectile of mass G and initial velocity Vo, then by applying Eq. [2.96] the final deflection of the beam can be estimated by
[2.97]
where the incipient collapse load of the beam,Ps = 4Mp/L given in Eq.
[2.43], is employed.
It should be noted that the above upper bound on the final displacement is based on a small deformation assumption. It may not provide an upper
Df Df o s
o s
o p
K P
GV P
GV L
£ + = = = M
2 2
2 8
Df Df o s
K
£ + = P
bound for the final displacement if a structure experiences a large defor- mation, when the geometric change and membrane forces become impor- tant. In fact, it has been demonstrated in Section 2.3.2 that the load-carrying capacity of the beam varies significantly with the deflection of the beam especially when the ends of the beam are axially constrained; thus the final deflection estimated by Eq. [2.97] would have a big error.
Initial loss in kinetic energy during collision
It has been pointed out in Section 2.4.2 that when two rigid-plastic bodies/structures collide with each other, there is an initial loss in the kinetic energy, denoted as Kloss. In this case, Eq. [2.95] will take the form
[2.98]
where the elastic strain energy is assumed to be negligible in the system, because both bodies/structures involved are taken as rigid-plastic. When applying the upper bound of the final displacement, the initial energy of the system,Koin Eq. [2.95], should be replaced by (Ko-Kloss).
The above initial loss in the kinetic energy can be avoided, if elastic- plastic structural models are used, or if an elastic-plastic contact spring is introduced between two rigid-plastic bodies/structures. A further discussion is given in Section 7.1.
Ein=Ko-Kloss=D
3
Dimensional analysis and experimental techniques
Dimensional analysis occupies an important place in engineering analysis. It forms the basis of small-scale model tests. This chapter introduces the concept and method of dimensional analysis with ref- erence to plastic collapse of structures, discusses the similitude requirements for model testing and presents several experimental techniques for studying the energy absorption of structures.